Mr Johnson uses an interview in The Sunday Telegraph to make a direct appeal to his party’s core values, presenting himself as a “tax-cutting Conservative” and promising to cut council tax if re-elected as London mayor.

His intervention will be seen as a gesture to his party’s leadership, which is facing growing claims that it is losing touch with grassroots supporters.

As well as his insistence on a low-tax agenda, Mr Johnson cites freedom, democracy and low government spending as his key beliefs, and vows to crack down further on crime.

Mr Johnson, who takes on Labour’s Ken Livingstone in this week’s mayoral election in London, does not mention issues such as gay marriage and the environment – policies that Mr Cameron has promoted recently, earning him criticism from his party.

The mayor also describes George Osborne, his possible future rival for the Tory leadership, as “the jaws of death” — a jibe that comes as the Chancellor faces continuing attacks over the Budget and last week’s confirmation that Britain was back in recession.

Mr Johnson makes his intervention as the Prime Minister struggles with a series of crises.

Downing Street last night signalled falling support for Jeremy Hunt, the Culture Secretary, who is accused of breaking the Ministerial Code in his dealings with Rupert Murdoch’s media empire.

A No 10 spokesman sent the first clear signal that Mr Hunt – who will submit details of his text messages and emails for scrutiny by Lord Justice Leveson’s inquiry into media standards this week – faces a separate Whitehall investigation into his conduct, which could lead to him being forced from Government.

The growing sense of crisis added to the Tories’ troubles ahead of Thursday’s local elections.

They are expected to lose hundreds of council seats, which would further hit grassroots morale. Last night a YouGov poll put the Tories on 29 per cent support 11 points behind Labour. The Liberal Democrats, with 11 per cent, were just above Ukip. Meanwhile:

Research by Lord Ashcroft, the former deputy Conservative chairman, shows that the party is out of touch with many ethnic minority voters.

Only four per cent of black voters said they identified with the Tories in a poll for Lord Ashcroft given to The Sunday Telegraph. The peer warns the party’s failure to woo black and Asian voters will exact a penalty in marginal seats at the next general election.

A survey by pollsters ComRes suggested Mr Cameron’s strong support for same-sex marriage could cost his party up to 30 seats in a general election. The UK Independence Party (Ukip) stands to benefit most from the drop in Tory support on this issue, the survey indicates, in what will be seen as further evidence that the Prime Minister is concentrating on policies that fail to appeal in his party’s heartlands.

The Sunday Telegraph has learnt the Treasury is “looking seriously” at a plan to retreat over plans to tax charitable donations by the wealthy – another policy attacked by traditional Tory supporters and backbench MPs.

The plan would see rich donors surrender some of their tax relief to charities. It would protect charities’ incomes but would still be unlikely to win the backing of philanthropists. The low-tax call from Mr Johnson follows last week’s intervention by Liam Fox.

The former defence secretary urged further cuts in state spending to make room for tax cuts aimed at business – a demand being made ever more volubly by the Tory Right.

The mayor sends a clear signal that he believes in a low-tax economy for all groups of voters, with the emphasis on lower council tax and pro-growth policies to help business.

“I certainly think London needs to be tax competitive,” Mr Johnson says. “I’ve got to look what I can do to bear down on people’s expenses … We have frozen council tax over four years, we’ll have cut it by 10 per cent in the next four years.”

He describes in vivid terms his “fight” to obtain the best deal for London.

“We’ve saved Crossrail [the new railway in London] from the jaws of death – aka George Osborne,” he says.

“I don’t mind having a row in Whitehall, I don’t mind how much plaster comes off the ceiling, providing it’s done in a robust and sensible way and London gets the money it needs.”

Asked what are the key values driving his campaign, Mr Johnson replies: “Freedom, democracy, taxpayer value and building up the sense of neighbourliness and duty towards each other — how’s that?”

He also echoes the Eurosceptic views of most Tory MPs by insisting “Parliament should be sovereign” with Britain better able to resist diktats from Europe.

He says, of the radical preacher wanted on terror charges in Jordan: “Abu Qatada should be put on the plane, there’s no doubt about it.”

Polls show Mr Johnson heading for victory over Mr Livingstone on Thursday but he insists the contest is “going down to the wire”.

In an attempt to deflect attention from the likelihood of his taking over from Mr Cameron as Tory leader, he says his chances are the same as his “being locked in a disused fridge”.

His race is the most high-profile contest in this week’s local elections across Britain, in which the Tories overall are expected to perform badly in the wake of crises hitting the Government.

Heavy defeats will add to the storm concerning Mr Hunt’s political future. This weekend, the Culture Secretary appears closer to being the subject of an inquiry by Sir Alex Allan, the independent adviser on the Ministerial Code, on his conduct over the bid by Mr Murdoch’s News Corporation for BSkyB last year.

The Culture Secretary’s special adviser, Adam Smith, resigned last week after emails published by the Leveson Inquiry appeared to show him passing confidential information about the bid to News Corp at a time when Mr Hunt had a quasi-judicial role in the process.

The No 10 spokesman added: “Once Jeremy Hunt’s evidence is made public and he is questioned [by Leveson], if there is anything that suggests there has been a breach of the code, the Prime Minister would, of course, act.”

Senior Tory and Lib Dem MPs have called for an independent inquiry.

However, Andrew Mitchell, the International Development Secretary, rallied round his cabinet colleague last night – although he too hinted at an inquiry.

“Jeremy is as straight as a die,” Mr Mitchell said. “He will shortly appear before the Leveson Inquiry where he will be cross-examined by a very tough barrister. Let people wait until he has done that before they judge him.

“I have no doubt that then, if questions remain, the Prime Minister will subject him to the full rigour of the Ministerial Code.”

Lord Ashcroft’s research, published in today’s Sunday Telegraph, shows that Mr Cameron’s party is out of touch with Britain’s growing ethnic minority groups, which now account for 10 per cent of the country’s population.

Just 29 per cent of more than 10,000 people polled by the former deputy Conservative chairman said they thought the Tories “understand minorities”.

Nearly 70 per cent of black voters polled said they thought Conservative politicians looked down on people from different ethnic or religious backgrounds more than those from other parties.

In a separate survey of more than 2,000 people, by ComRes, Mr Cameron’s support for changing the law to permit same-sex marriages was identified as high-risk in terms of lost votes for his party.

If this week’s local elections were a general election, the poll, commissioned by the Coalition for Marriage pressure group, suggested that the Tories could lose between eight and 30 seats over the issue.

Andrew Hawkins, the chief executive of ComRes, said the Tories were losing three voters for every one they gained on the issue.

Ukip, which is expected to make gains in this week’s local elections, would be the main beneficiary, the survey showed.

Last night, it was disclosed that Teresa Gorman, the former Tory MP who rebelled against John Major’s government in the 1990s, had defected to Ukip. And Mr Osborne, the Chancellor, is this weekend facing opposition from within the Cabinet on another policy announced in last month’s Budget.

Iain Duncan Smith, the Work and Pensions Secretary, is fighting the Chancellor’s plan to cut welfare spending by an extra £10 billion between 2014 and 2016.

When asked if he considered such a cut acceptable, Mr Duncan Smith said: “My view is it’s not … There is in my view no such thing as an easy cut to welfare. Some people think there is: until I show them where we spend the money.”

His comments fuel the sense of a botched Budget.

Separately, David Gauke, the Tax Minister, will meet charities this week in a bid to end the row over plans to cap tax relief for wealthy philanthropists.

The Chancellor is “looking seriously” at a plan proposed by the British Red Cross, whereby charities would be allowed to keep the tax relief currently enjoyed by their richest donors.

More than 800 charities are calling for the abandonment of the plan announced last month to limit the tax relief available to large donors at £50,000 a year or 25 per cent of a donor’s annual earnings – whichever is greater.

Some wealthy givers have said they will be less generous if the current proposals remain.

Under the new plan, wealthy donors could opt to surrender tax relief on their donations, transferring into the charity they are giving to.

However, if the donor still chose to receive the tax relief themselves it would be subject to the annual limits proposed by the Chancellor in the Budget.

The plan has won backing from the Institute for Fundraising, which represents hundreds of charities including Oxfam, the NSPCC and Cancer Research.